Locally convex spaces and Schur type properties

Gabriyelyan, Saak

doi:10.5186/aasfm.2019.4417

Cited by 17 publications

(29 citation statements)

References 21 publications

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“…If E is a locally convex space, we denote by Proof. The clauses (i)-(iii) are equivalent by Proposition 2.5, and Theorem 1.2 of [19] exactly states that (i) and (iv) are equivalent.…”

Section: The Josefson-nissenzweig Property In Locally Convex Spacesmentioning

confidence: 82%

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

Банах

Ka̧kol

Śliwa

2019

RACSAM

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We define a locally convex space E to have the Josefson-Nissenzweig property (JNP) if the identity map (E ′ , σ(E ′ , E)) → (E ′ , β * (E ′ , E)) is not sequentially continuous. By the classical Josefson-Nissenzweig theorem, every infinite-dimensional Banach space has the JNP.We show that for a Tychonoff space X, the function space Cp(X) has the JNP iff there is a weak * null-sequence {µn}n∈ω of finitely supported sign-measures on X with unit norm. However, for every Tychonoff space X, neither the space B1(X) of Baire-1 functions on X nor the free locally convex space L(X) over X has the JNP. We also define two modifications of the JNP, called the universal JNP and the JNP everywhere (briefly, the uJNP and eJNP), and thoroughly study them in the classes of locally convex spaces, Banach spaces and function spaces. We provide a characterization of the JNP in terms of operators into locally convex spaces with the uJNP or eJNP and give numerous examples clarifying relationships between the considered notions.2010 Mathematics Subject Classification. Primary 46A03; Secondary 46E10, 46E15.

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“…If E is a locally convex space, we denote by Proof. The clauses (i)-(iii) are equivalent by Proposition 2.5, and Theorem 1.2 of [19] exactly states that (i) and (iv) are equivalent.…”

Section: The Josefson-nissenzweig Property In Locally Convex Spacesmentioning

confidence: 82%

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

Банах

Ka̧kol

Śliwa

2019

RACSAM

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“…Then L(X) is the direct sum of |X|-many copies of R, and hence the strong dual of L(X) is R |X| . Since R |X| has the Schur property, Proposition 3.1 of [13] implies that L(X) has the (sDP )-property.…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

“…He used this result to show that every Banach space C(K) has the (DP ) property, see [18,Théorème 1]. Extending this result to locally convex spaces and following [13], we consider the following "sequential" version of the (DP ) property: an lcs E is said to have the sequential Dunford-Pettis property ((sDP ) property) if given weakly null sequences {x n } n∈N and {χ n } n∈N in E and the strong dual E ′ β of E, respectively, then lim n χ n (x n ) = 0. In [15], we showed that C p (X) has the (sDP ) property for every Tychonoff space X.…”

Section: Introductionmentioning

confidence: 99%

Locally convex properties of free locally convex spaces

Gabriyelyan

2019

Journal of Mathematical Analysis and Applications

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Let L(X) be the free locally convex space over a Tychonoff space X. We show that the following assertions are equivalent:We prove that L(X) is a (DF )-space iff X is a countable discrete space. We show that there is a countable Tychonoff space X such that L(X) is a quasi-(DF )-space but is not a c0-quasibarrelled space. For each non-metrizable compact space K, the space L(K) is a (df )-space but is not a quasi-(DF )-space. If X is a µ-space, then L(X) has the Grothendieck property iff every compact subset of X is finite. We show that L(X) has the Dunford-Pettis property for every Tychonoff space X. If X is a sequential µ-space (for example, metrizable), then L(X) has the sequential Dunford-Pettis property iff X is discrete.2000 Mathematics Subject Classification. Primary 46A03, 46A08; Secondary 54C35.

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“…A subset A of a topological space X is called sequentially compact if every sequence in A contains a subsequence which is convergent in A. Following [21], a Tychonoff space X is sequentially angelic if a subset K of X is compact if and only if K is sequentially compact. It is clear that every angelic space is sequentially angelic.…”

Section: The (Dp ) Property and The Grothendieck Property For C P (X)mentioning

confidence: 99%

“…He used this result to show that every Banach space C(K) has the (DP ) property, see [23,Théorème 1]. Extending this result to locally convex spaces (lcs, for short) and following [21], we consider the following "sequential" version of the (DP ) property. Definition 1.2.…”

Section: Introductionmentioning

confidence: 99%

Dunford–Pettis type properties and the Grothendieck property for function spaces

Gabriyelyan

Ka̧kol

2019

Rev Mat Complut

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For a Tychonoff space X, let C k (X) and Cp(X) be the spaces of real-valued continuous functions C(X) on X endowed with the compact-open topology and the pointwise topology, respectively. If X is compact, the classic result of A. Grothendieck states that C k (X) has the Dunford-Pettis property and the sequential Dunford-Pettis property. We extend Grothendieck's result by showing that C k (X) has both the Dunford-Pettis property and the sequential Dunford-Pettis property if X satisfies one of the following conditions: (i) X is a hemicompact space, (ii) X is a cosmic space (=a continuous image of a separable metrizable space), (iii) X is the ordinal space [0, κ) for some ordinal κ, or (vi) X is a locally compact paracompact space. We show that if X is a cosmic space, then C k (X) has the Grothendieck property if and only if every functionally bounded subset of X is finite. We prove that Cp(X) has the Dunford-Pettis property and the sequential Dunford-Pettis property for every Tychonoff space X, and Cp(X) has the Grothendieck property if and only if every functionally bounded subset of X is finite.2000 Mathematics Subject Classification. Primary 46A03; Secondary 54H11. Key words and phrases. function space, Dunford-Pettis property, Grothendieck property. Jerzy Kakol gratefully acknowledges the nancial support he received from the Center for Advanced Studies in Mathematics of the Ben Gurion University of the Negev during his visit April, 2018.

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Locally convex spaces and Schur type properties

Cited by 17 publications

References 21 publications

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

Locally convex properties of free locally convex spaces

Dunford–Pettis type properties and the Grothendieck property for function spaces

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